P-adaptive Hermite methods for initial value problems^∗

¹ Department of Mathematics and Statistics, The University of New Mexico, MSC03 2150, Albuquerque, 87131-0001 NM, USA
xqroy@math.unm.edu
² Department of Mathematics, Southern Methodist University, PO Box 750156, Dallas, 75275-0156 TX, USA
thagstrom@smu.edu

Received: 29 September 2009

Abstract

We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems posed in 1+1 and 2+1 dimensions.